The Annals of Probability

A Gaussian kinematic formula

Jonathan E. Taylor

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Abstract

In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl–Steiner tube formulae and the Chern–Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold M. Specifically, we consider random fields of the form fp=F(y1(p),…,yk(p)) for FC2(ℝk;ℝ) and (y1,…,yk) a vector of C2 i.i.d. centered, unit-variance Gaussian fields.

The analogue of the Weyl–Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest.

As in the classical Weyl–Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, χ, of the excursion sets Mf−1[u,+∞)=My−1(F−1[u,+∞)) of the field f.

The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.

Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 122-158.

Dates
First available in Project Euclid: 17 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1140191534

Digital Object Identifier
doi:10.1214/009117905000000594

Mathematical Reviews number (MathSciNet)
MR2206344

Zentralblatt MATH identifier
1094.60025

Subjects
Primary: 60G15: Gaussian processes 60G60: Random fields 53A17: Kinematics 58A05: Differentiable manifolds, foundations
Secondary: 60G17: Sample path properties 62M40: Random fields; image analysis 60G70: Extreme value theory; extremal processes

Keywords
Random fields Gaussian processes manifolds Euler characteristic excursions Riemannian geometry

Citation

Taylor, Jonathan E. A Gaussian kinematic formula. Ann. Probab. 34 (2006), no. 1, 122--158. doi:10.1214/009117905000000594. https://projecteuclid.org/euclid.aop/1140191534


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